163 research outputs found
Compactly Supported Tensor Product Complex Tight Framelets with Directionality
Although tensor product real-valued wavelets have been successfully applied
to many high-dimensional problems, they can only capture well edge
singularities along the coordinate axis directions. As an alternative and
improvement of tensor product real-valued wavelets and dual tree complex
wavelet transform, recently tensor product complex tight framelets with
increasing directionality have been introduced in [8] and applied to image
denoising in [13]. Despite several desirable properties, the directional tensor
product complex tight framelets constructed in [8,13] are bandlimited and do
not have compact support in the space/time domain. Since compactly supported
wavelets and framelets are of great interest and importance in both theory and
application, it remains as an unsolved problem whether there exist compactly
supported tensor product complex tight framelets with directionality. In this
paper, we shall satisfactorily answer this question by proving a theoretical
result on directionality of tight framelets and by introducing an algorithm to
construct compactly supported complex tight framelets with directionality. Our
examples show that compactly supported complex tight framelets with
directionality can be easily derived from any given eligible low-pass filters
and refinable functions. Several examples of compactly supported tensor product
complex tight framelets with directionality have been presented
Nonhomogeneous Wavelet Systems in High Dimensions
It is of interest to study a wavelet system with a minimum number of
generators. It has been showed by X. Dai, D. R. Larson, and D. M. Speegle in
[11] that for any real-valued expansive matrix M, a homogeneous
orthonormal M-wavelet basis can be generated by a single wavelet function. On
the other hand, it has been demonstrated in [21] that nonhomogeneous wavelet
systems, though much less studied in the literature, play a fundamental role in
wavelet analysis and naturally link many aspects of wavelet analysis together.
In this paper, we are interested in nonhomogeneous wavelet systems in high
dimensions with a minimum number of generators. As we shall see in this paper,
a nonhomogeneous wavelet system naturally leads to a homogeneous wavelet system
with almost all properties preserved. We also show that a nonredundant
nonhomogeneous wavelet system is naturally connected to refinable structures
and has a fixed number of wavelet generators. Consequently, it is often
impossible for a nonhomogeneous orthonormal wavelet basis to have a single
wavelet generator. However, for redundant nonhomogeneous wavelet systems, we
show that for any real-valued expansive matrix M, we can always
construct a nonhomogeneous smooth tight M-wavelet frame in with a
single wavelet generator whose Fourier transform is a compactly supported
function. Moreover, such nonhomogeneous tight wavelet frames are
associated with filter banks and can be modified to achieve directionality in
high dimensions. Our analysis of nonhomogeneous wavelet systems employs a
notion of frequency-based nonhomogeneous wavelet systems in the distribution
space. Such a notion allows us to separate the perfect reconstruction property
of a wavelet system from its stability in function spaces
Modular frames for Hilbert C*-modules and symmetric approximation of frames
We give a comprehensive introduction to a general modular frame construction
in Hilbert C*-modules and to related modular operators on them. The Hilbert
space situation appears as a special case. The reported investigations rely on
the idea of geometric dilation to standard Hilbert C*-modulesover unital
C*-algebras that admit an orthonormal Riesz basis. Interrelations and
applications to classical linear frame theory are indicated. As an application
we describe the nature of families of operators {S_i} such that SUM_i
S*_iS_i=id_H, where H is a Hilbert space. Resorting to frames in Hilbert spaces
we discuss some measures for pairs of frames to be close to one another. Most
of the measures are expressed in terms of norm-distances of different kinds of
frame operators. In particular, the existence and uniqueness of the closest
(normalized) tight frame to a given frame is investigated. For Riesz bases with
certain restrictions the set of closetst tight frames often contains a multiple
of its symmetric orthogonalization (i.e. L\"owdin orthogonalization).Comment: SPIE's Annual Meeting, Session 4119: Wavelets in Signal and Image
Processing; San Diego, CA, U.S.A., July 30 - August 4, 2000. to appear in:
Proceedings of SPIE v. 4119(2000), 12 p
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